A Laplace Duality for Integration

We consider the integral $v\text {(}y\text {)}=\int _{K_{y}}f\text {(}\mathbf {x}\text {)}d\mathbf {x}$ on a domain $K_{y}=\{\mathbf {x}\in \mathbb {R}^{d}~:~g\text {(}\mathbf {x}\text {)}\leq y\}$ , where g is nonnegative and $K_{y}$ is compact for all $y\in [0,+\infty \text {)}$ . Under some assumptions, we show that for every $y\in \text {(}0,\infty \text {)}$ there exists a distinguished scalar $\lambda _{y}\in \text {(}0,+\infty \text {)}$ such that $v\text {(}y\text {)}=\int _{\mathbb {R}^{d}}f\text {(}\mathbf {x}\text {)}\exp \text {(} - \lambda _{y}\,g\text {(}\mathbf {x}\text {)}\text {)}\,d\mathbf {x}$ , which is the counterpart analogue for integration of Lagrangian duality for optimization. A crucial ingredient is the Laplace transform, the analogue for integration of Legendre-Fenchel transform in optimization. In particular, if both f and g are positively homogeneous then $\lambda _{y}$ is a simple explicitly rational function of y. In addition if g is quadratic form then computing $v\text {(}y\text {)}$ reduces to computing the integral of f with respect to a specific Gaussian measure for which exact and approximate numerical methods (e.g. cubatures) are available.